All games must be played mechanically if there is no free choice between them. The final number of steps in the game is less than an infinite number of events in the past.

You are the one that gave the analogy of a chess game where there are a finite set of moves. If you believe in a "subject of free choice" then that means I am not bound by the rules of the game (chess) and thus can move my pieces any way I want to to and change the rules of the game whenever I feel like it.

Make up your mind: finite number of moves (no free agency) infinite number of moves (free agency).

Theravada _totally_ denies the existence of sattа (_totally_ denies the existence of the subject of free choice).

You keep saying that over and again but: 1. You have not provided any proof of this. 2. You have not provided any proof of Mahayana supporting the idea of a "free agent". Mahayana Abhidharma is not that different to Theravada. Have you studied it?

"My religion is not deceiving myself."Jetsun Milarepa 1052-1135 CE

"Butchers, prostitutes, those guilty of the five most heinous crimes, outcasts, the underprivileged: all are utterly the substance of existence and nothing other than total bliss."The Supreme Source - The Kunjed Gyalpo
The Fundamental Tantra of Dzogchen Semde

You keep saying that over and again but: 1. You have not provided any proof of this. 2. You have not provided any proof of Mahayana supporting the idea of a "free agent". Mahayana Abhidharma is not that different to Theravada. Have you studied it?

Vasubandhu’s pudgala is the five skandhas (which exist) - not something non-existent.

The Path of Purification p. 233-235
...
Bhikkhu Ñánamoli:
...
It then gives six kinds of paññatti “according to the commentarial method but not
in the texts”:

(1) Concept of the existent (vijjamána-paññatti), which is the conceptualizing of (making known) a dhamma that is existent, actual, become, in the true and ultimate sense (e.g. aggregates, etc.).

(2) Concept of the non-existent, which is, for example, the conceptualizing of “female,” “male,” “persons,” etc., which are non-existent by that standard and are only established by means of current speech in the world; similarly “such impossibilities as concepts of a fifth truth or the other sectarians’ Atom, Primordial Essence, World Soul, and the like.”
...
(6) Concept of the non-existent based on the non-existent, e.g. “banker’s son,” both being non-existent.

Buddha Abhidhamma Ultimate Science
by Dr. Mehm Tin Mon

p. 362-363
Six kinds of Sadda-pannatti

1 Vijjamàna-pannatti (real concept)
When a name is given to something which exists in reality, then that name is called ‘vijjamàna-pannatti’. All the names of the ultimate realities (paramatthas) belong to this class; eg., Råpa, citta, cetasika, vedanà, sannà, vitakka.

2 Avijjamàna-pannatti (unreal concept)
When a name is designated to something which does not exist in reality, then that name is called ‘avijjamàna-pannatti’. All the names of things which are not ultimate realities belong to this class; eg., Man, dog, house, school, hill, cave.
...
6 Avijjamànena-avijjamàna-pannatti (unreal and unreal concept) It is a sadda-pannatti which makes known a compound name formed by combining and unreal concept with an unreal concept. eg., Ràja-putta (king’s son), movie-actress, company-director, head-master.

You keep saying that over and again but: 1. You have not provided any proof of this. 2. You have not provided any proof of Mahayana supporting the idea of a "free agent". Mahayana Abhidharma is not that different to Theravada. Have you studied it?

Vasubandhu’s pudgala is the five skandhas (which exist) - not something non-existent.

The Path of Purification p. 233-235
...
Bhikkhu Ñánamoli:
...
It then gives six kinds of paññatti “according to the commentarial method but not
in the texts”:

(1) Concept of the existent (vijjamána-paññatti), which is the conceptualizing of (making known) a dhamma that is existent, actual, become, in the true and ultimate sense (e.g. aggregates, etc.).

(2) Concept of the non-existent, which is, for example, the conceptualizing of “female,” “male,” “persons,” etc., which are non-existent by that standard and are only established by means of current speech in the world; similarly “such impossibilities as concepts of a fifth truth or the other sectarians’ Atom, Primordial Essence, World Soul, and the like.”
...
(6) Concept of the non-existent based on the non-existent, e.g. “banker’s son,” both being non-existent.

Buddha Abhidhamma Ultimate Science
by Dr. Mehm Tin Mon

p. 362-363
Six kinds of Sadda-pannatti

1 Vijjamàna-pannatti (real concept)
When a name is given to something which exists in reality, then that name is called ‘vijjamàna-pannatti’. All the names of the ultimate realities (paramatthas) belong to this class; eg., Råpa, citta, cetasika, vedanà, sannà, vitakka.

2 Avijjamàna-pannatti (unreal concept)
When a name is designated to something which does not exist in reality, then that name is called ‘avijjamàna-pannatti’. All the names of things which are not ultimate realities belong to this class; eg., Man, dog, house, school, hill, cave.
...
6 Avijjamànena-avijjamàna-pannatti (unreal and unreal concept) It is a sadda-pannatti which makes known a compound name formed by combining and unreal concept with an unreal concept. eg., Ràja-putta (king’s son), movie-actress, company-director, head-master.

Mahayana Abhidharma does not differ.

Anatman, you know?

"My religion is not deceiving myself."Jetsun Milarepa 1052-1135 CE

"Butchers, prostitutes, those guilty of the five most heinous crimes, outcasts, the underprivileged: all are utterly the substance of existence and nothing other than total bliss."The Supreme Source - The Kunjed Gyalpo
The Fundamental Tantra of Dzogchen Semde

Well, see, the thing is that due to ignorance (and ensuing rebirth) we actually play the same game over and over, thus denying ourselves the opportunity of playing a winning game.

Yup the possibility of semi-stable loops of events was not accounted for. The reasoning can also be twisted around as follows:

Enlightenment occurs but once. In infinite time the number of unenlightened lives is infinite. Therefore the probability of enlightenment is zero.]

Most people when confronted with this sort of thing do not investigate their conceptions of infinity at all, do not do any meaningful research into the peculiar things infinities do to our usual reasoning methods, and then simply revert to mindlessly re-asserting whatever pet theory appeals to them at the moment. Such is samsara.

Vasubandhu’s pudgala is the five skandhas (which exist) - not something non-existent.

So your issue with the Theravada, Abhidhamma in particular is the anatta doctrine and where you believe that this means beings don't exist? And you believe this issue doesn't exist in Mahayana? Mahayana is a very big school but the anatman doctrine is found in all of them, although perhaps to a lesser degree (i.e., not necessarily non-existence) in some schools. And then some have argued that the non-existence - extreme anatta is not found in all Theravada sub-traditions, for example some (or most?) forest traditions.

Vasubandhu’s pudgala is the five skandhas (which exist) - not something non-existent.

So your issue with the Theravada, Abhidhamma in particular is the anatta doctrine and where you believe that this means beings don't exist? And you believe this issue doesn't exist in Mahayana? Mahayana is a very big school but the anatman doctrine is found in all of them, although perhaps to a lesser degree (i.e., not necessarily non-existence) in some schools. And then some have argued that the non-existence - extreme anatta is not found in all Theravada sub-traditions, for example some (or most?) forest traditions.

I agree. I do believe that his issue is essentially an aversion to the BUDDHIST notion of anatma(n) rather than anything specific about Theravada.

"My religion is not deceiving myself."Jetsun Milarepa 1052-1135 CE

"Butchers, prostitutes, those guilty of the five most heinous crimes, outcasts, the underprivileged: all are utterly the substance of existence and nothing other than total bliss."The Supreme Source - The Kunjed Gyalpo
The Fundamental Tantra of Dzogchen Semde

The infinite monkey theorem: “The probability that an infinite number of monkeys will print any given text on the first attempt is 1”. Here a "monkey" is the past life. A "text" is a sequence of combinations of dhammas, culminating in the realization of Nibbana. As the number of past lives is not limited, the probability of nibbanization in past lives is 1.

There is no 'infinite monkey theorem' - it's just an internet meme. Put a monkey near a typewriter, and it won't type. It would in all likelihood ignore it. Train it to punch keys - sure. But then you might just as well build an algorithm that enters random strings of characters. And I don't believe that such an algorithm would ever produce a meaningful sentence, let alone a story. The OP is a meaningless thought-experiment - and I don't say that out of ideological commitment to Buddhism, but because it doesn't do justice to its existential nature, the actual living dynamics of Buddhist life and practice. Rather it treats it as an abstraction and then claims that it doesn't make sense.

Vasubandhu’s pudgala is the five skandhas (which exist) - not something non-existent.

So your issue with the Theravada, Abhidhamma in particular is the anatta doctrine and where you believe that this means beings don't exist? And you believe this issue doesn't exist in Mahayana? Mahayana is a very big school but the anatman doctrine is found in all of them, although perhaps to a lesser degree (i.e., not necessarily non-existence) in some schools. And then some have argued that the non-existence - extreme anatta is not found in all Theravada sub-traditions, for example some (or most?) forest traditions.

I agree. I do believe that his issue is essentially an aversion to the BUDDHIST notion of anatma(n) rather than anything specific about Theravada.

I think so. I was trying to imagine if the argument would be any different with or without the Mayahana view of shunyata and...you end up in exactly the same place, because (a) anatman as posited in Theraveda is not disputed by the Mahayana schools, (b) all schools are equally imprecise on the finite or infinite nature of living beings and cyclic time.

This thought experiment, though interesting, does commit a basic error in that it's taking concepts of infinite numbers from modern mathematics and imposing them on pre-modern philosophy. As catmoon commented, transfinite numbers do not behave in intuitive ways.

A test is a combination of causes and conditions leading to an outcome. The outcome is incidental if the test leading to any outcome has other outcomes.

A random event is any collection of random outcomes.

A deterministic event is an outcome that is sure to occur during a given test.

Impossible event is the outcome, which obviously does not happen during this test.

So, the test is the previous combination of a finite number of dhammas. The end result is a subsequent combination of a finite number of dhammas.

In the absence of a subject of free choice, all events are reduced to random, deterministic and impossible (just as in the physical model).

If there are random events, the topic about the infinite monkeys theorem is valid. If there are no random events, then all events are reduced to deterministic and impossible. If there are no random events, then all possible events are deterministic.

The final combination of dhammas before the manifestation of Nibbana is deterministic event А.

The objection is reduced to the statement that a deterministic event can not occur in an infinite number of tests.

This is possible only if in this infinite set of tests there is no combination of causes and conditions that precedes such a deterministic event А. But this combination - like any test - is also a deterministic event.

If there is no test, there are no reasons for it with conditions, and so we deny the causes and conditions to infinity. In all infinity of causes and conditions, there is not a single combination that could trigger a sequence of steps that ends with a given deterministic event А.

The final combination of dhammas before the manifestation of Nibbana is an impossible event.

A test is a combination of causes and conditions leading to an outcome. The outcome is incidental if the test leading to any outcome has other outcomes.

A random event is any collection of random outcomes.

A deterministic event is an outcome that is sure to occur during a given test.

Impossible event is the outcome, which obviously does not happen during this test.

So, the test is the previous combination of a finite number of dhammas. The end result is a subsequent combination of a finite number of dhammas.

In the absence of a subject of free choice, all events are reduced to random, deterministic and impossible (just as in the physical model).

If there are random events, the topic about the infinite monkeys theorem is valid. If there are no random events, then all events are reduced to deterministic and impossible. If there are no random events, then all possible events are deterministic.

The final combination of dhammas before the manifestation of Nibbana is deterministic event А.

The objection is reduced to the statement that a deterministic event can not occur in an infinite number of tests.

This is possible only if in this infinite set of tests there is no combination of causes and conditions that precedes such a deterministic event А. But this combination - like any test - is also a deterministic event.

If there is no test, there are no reasons for it with conditions, and so we deny the causes and conditions to infinity. In all infinity of causes and conditions, there is not a single combination that could trigger a sequence of steps that ends with a given deterministic event А.

The final combination of dhammas before the manifestation of Nibbana is an impossible event.

If i understand what you say above, you are arguing that Nibbana is an outcome, meaning it is the product of a combination of causes and conditions, correct?

Mod note: this is a pointless argument about a completely artificial ‘thought experiment’ which has no relevance to Buddhist principles, thought and practice. It might be noted that central to these are the ‘four immeasurables’ which, as the name implies, are not subject to quantitative evaluation and hence totally outside the scope of any such calculations. Accordingly, this thread is deemed to have run its course, as its originator keeps repeating the same arguments ad nauseum without any indication of ‘arguing in good faith’.